A Study of Braids: (484 Mathematics and Its Applications)

This book provides a comprehensive exposition of the theory of braids, beginning with the basic mathematical definitions and structures. Among the many topics explained in detail are: the braid group for various surfaces; the solution of the word problem for the braid group; braids in the context of knots and links (Alexander's theorem); Markov's theorem and its use in obtaining braid invariants; the connection between the Platonic solids (regular polyhedra) and braids; the use of braids in the solution of algebraic equations. Dirac's problem and special types of braids termed Mexican plaits are also discussed. Audience: Since the book relies on concepts and techniques from algebra and topology, the authors also provide a couple of appendices that cover the necessary material from these two branches of mathematics. Hence, the book is accessible not only to mathematicians but also to anybody who might have an interest in the theory of braids. In particular, as more and more applications of braid theory are found outside the realm of mathematics, this book is ideal for any physicist, chemist or biologist who would like to understand the mathematics of braids. With its use of numerous figures to explain clearly the mathematics, and exercises to solidify the understanding, this book may also be used as a textbook for a course on knots and braids, or as a supplementary textbook for a course on topology or algebra.

Table of Contents:
1. Introduction & Foundations.- 1. Various types of braids.- 2. A definition of a braid.- 3. An elementary move and braid equivalence.- 4. Braid projection.- 5. Braid permutation, pure braid.- 2. The Braid Group.- 1. Defunition of the braid group.- 2. A presentation for the braid group.- 3. The completeness of the relations.- 4. Elementary properties of the braid group.- 5. A braid invariant.- 3. Word Problem.- 1. Word problem for the braid group.- 2. A solution of the word problem.- 3. A presentation for the pure n-braid group.- 4. Special types of braids.- 1. Mexicar plaits.- 2. Generators of the Mexican plaits.- 3. An algorithm for Mexican plaits.- 4. Examples of the use of the algorithm.- 5. Quotient groups of the braid group.- 1. Sywumetric group and the braid group.- 2. Platoric solids and quotient groups of Bn.- 3. Finite quotient groups of B3.- 4. The firite quotient group B4(3).- 5. The finite quotient group B5(3).- 6. Isotopy of braids.- 1. Equivalence and isotopy.- 2. Words.- 3. Several interpretations of equivalence.- 4. Milnor invariant.- 7. Homotopy braid theory.- 1. Homotopy.- 2. Tangles and homotopy.- 3. Homotopy braid group.- 4. Homotopy braid invariants.- 5. Tangles and braids.- 8. Grom knots to braids.- 1. Knot theory — a quick review.- 2. Quasi-braids.- 3. Braided links.- 4. Alexander’s theorem.- 5. Knot invariants via braid invariants.- 9. Markov’s theorem.- 1. A theorem due to Markov.- 2. Proof of Markov’s theorem — I.- 3. Proof of Markov’s theorem — II.- 4. Applications.- 10. Knot invariants.- 1. Burau representation.- 2. Alexander polynomial.- 3. Jones polynomial.- 4. Alexander versus Jones.- 11. Braid groups on surfaces.- 1. Divac’s Problem.- 2. Braid group on S2.- 3. Braid group on the surface F.- 4. Braid group on P2.- 5. Braidgroup on T2.- 6. Word problem for Bn(S2).- 12. Algebraic equations.- 1. Configuration spaces.- 2. Complete solvability.- Appendix I — Group theory.- 1. Equivalence relation.- 2. Groups and a bit of ring theory.- 3. Free group.- 4. Presentations of groups.- 5. Word problem.- 6. Reidemeister-Schreier method, presentation of a subgroup.- 7. Triangle groups.- Appendix II — Topology.- 1. Fundamental concepts of Topology.- 2. Homotopy.- 3. Fundamental group.- 4. Manifolds.- Appendix III — Symplectic group.- 1. Symplectic group.- Appendix IV.- Appendix V.